Autoregressive Model

This example shows how to compute and plot the impulse response function for an autoregressive (AR) model. The AR(p) model is given by

yt=μ+ϕ(L)-1εt,

where ϕ(L) is a p-degree AR operator polynomial, (1-ϕ1L-…-ϕpLp).

An AR process is stationary provided that the AR operator polynomial is stable, meaning all its roots lie outside the unit circle. In this case, the infinite-degree inverse polynomial, ψ(L)=ϕ(L)-1, has absolutely summable coefficients, and the impulse response function decays to zero.

ARMA Model

This example shows how to plot the impulse response function for an autoregressive moving average (ARMA) model. The ARMA(p, q) model is given by

yt=μ+θ(L)ϕ(L)εt,

where θ(L) is a q-degree MA operator polynomial, (1+θ1L+…+θqLq), and ϕ(L) is a p-degree AR operator polynomial, (1-ϕ1L-…-ϕpLp).

An ARMA process is stationary provided that the AR operator polynomial is stable, meaning all its roots lie outside the unit circle. In this case, the infinite-degree inverse polynomial, ψ(L)=θ(L)/ϕ(L) , has absolutely summable coefficients, and the impulse response function decays to zero.